What is the ruling on teaching the Qur'an through games, and does this fall under Allah's saying: (No remembrance comes to them from their Lord as a fresh admonition but they listen to it while they play)?

The link referred to is an educational link for children, enabling them to memorize the Holy Quran by reordering its verses that are displayed out of sequence. This educational method is not among the blameworthy "play" mentioned in the Quran. Rather, it is a means to convey the truths of the Quran to the young generation, relying on the combination of the senses of hearing, sight, touch, and movement. The blameworthy play involves turning away and being heedless of the message of truth. Furthermore, the jurisprudential rule states, "Means take the rulings of their ends," and the end here is the memorization of the Holy Quran.

The link consists of "tests" and reordering of verses, which is permissible and there is no harm in it.